mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.SetTheory.Cardinal.Subfield

Goal state before:
pp: case e'_4
α : Type u
s : Set α
inst✝ : DivisionRing α
⊢ max #↑s ℵ₀ = max (lift.{u, 0} #(Fin 6) + lift.{0, u} #↑s) ℵ₀

Tactic:
lift_uzero

Goal state after:
case e'_4
α : Type u
s : Set α
inst✝ : DivisionRing α
⊢ max #↑s ℵ₀ = max (lift.{u, 0} #(Fin 6) + #↑s) ℵ₀
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: R : Type uR
M : Type uM
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : Finite ι
i j : ι
⊢ (b.dualBasis i) (b j) = if j = i then 1 else 0

Tactic:
by
  convert! b.toDual_apply i j using 2
  rw [@eq_comm _ j i]

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
M' : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup M'
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : HasRankNullity.{u, u_1} R
f : M →ₗ[R] M'
⊢ lift.{u, v} (Module.rank R ↥f.range) + lift.{v, u} (Module.rank R ↥f.ker) = lift.{v, u} (Module.rank R M)

Tactic:
have := fun p : Submodule R M => Classical.decEq (M ⧸ p)
  rw [← f.quotKerEquivRange.lift_rank_eq, ← lift_add, rank_quotient_add_rank]

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
M' : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup M'
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : HasRankNullity.{v, u_1} R
f : M →ₗ[R] M'
p : Submodule R M
⊢ p ≤ comap f.rangeRestrict (Submodule.map f.rangeRestrict p)

Tactic:
rw [comap_map_eq]

Goal state after:
R : Type u_1
M : Type u
M' : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup M'
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : HasRankNullity.{v, u_1} R
f : M →ₗ[R] M'
p : Submodule R M
⊢ p ≤ p ⊔ f.rangeRestrict.ker
Module: Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n m : ℕ
f : Matrix (Fin m) (Fin n) R
g : Matrix (Fin n) (Fin m) R
⊢ opEquiv.mapMatrix.symm (opEquiv.mapMatrix g * opEquiv.mapMatrix f) = opEquiv.mapMatrix.symm 1 →
      opEquiv.mapMatrix.symm (opEquiv.mapMatrix f * opEquiv.mapMatrix g) = opEquiv.mapMatrix.symm 1 → n = m ↔
    (transposeAddEquiv (Fin m) (Fin n) R).toEquiv f * (transposeAddEquiv (Fin n) (Fin m) R).toEquiv g = 1 →
      (transposeAddEquiv (Fin n) (Fin m) R).toEquiv g * (transposeAddEquiv (Fin m) (Fin n) R).toEquiv f = 1 → n = m

Tactic:
rw [← (transposeAddEquiv ..).injective.eq_iff, ← (transposeAddEquiv (Fin m) ..).injective.eq_iff]

Goal state after:
R : Type u_1
inst✝ : Semiring R
n m : ℕ
f : Matrix (Fin m) (Fin n) R
g : Matrix (Fin n) (Fin m) R
⊢ (transposeAddEquiv (Fin n) (Fin n) R) (opEquiv.mapMatrix.symm (opEquiv.mapMatrix g * opEquiv.mapMatrix f)) =
        (transposeAddEquiv (Fin n) (Fin n) R) (opEquiv.mapMatrix.symm 1) →
      (transposeAddEquiv (Fin m) (Fin m) R) (opEquiv.mapMatrix.symm (opEquiv.mapMatrix f * opEquiv.mapMatrix g)) =
          (transposeAddEquiv (Fin m) (Fin m) R) (opEquiv.mapMatrix.symm 1) →
        n = m ↔
    (transposeAddEquiv (Fin m) (Fin n) R).toEquiv f * (transposeAddEquiv (Fin n) (Fin m) R).toEquiv g = 1 →
      (transposeAddEquiv (Fin n) (Fin m) R).toEquiv g * (transposeAddEquiv (Fin m) (Fin n) R).toEquiv f = 1 → n = m
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
s t : Submodule K V
hdisjoint : Disjoint s t
h_finrank_inf : finrank K ↥(s ⊓ t) = 0
hdim : finrank K V = finrank K ↥s + finrank K ↥t
⊢ finrank K ↥(s ⊔ t) = finrank K ↥(s ⊔ t) + finrank K ↥(s ⊓ t)

Tactic:
h_finrank_inf,

Goal state after:
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
s t : Submodule K V
hdisjoint : Disjoint s t
h_finrank_inf : finrank K ↥(s ⊓ t) = 0
hdim : finrank K V = finrank K ↥s + finrank K ↥t
⊢ finrank K ↥(s ⊔ t) = finrank K ↥(s ⊔ t) + 0
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
p : Submodule K V
v : V
hv : v ∉ p
⊢ p ⊓ K ∙ v ≤ ⊥

Tactic:
intro x

Goal state after:
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
p : Submodule K V
v : V
hv : v ∉ p
x : V
⊢ x ∈ p ⊓ K ∙ v → x ∈ ⊥
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ Y : C
p₁ : ExactPairing X₁ Y
p₂ : ExactPairing X₂ Y
⊢ ((ᘁ𝟙 Y) ≫ ᘁ𝟙 Y) = 𝟙 X₁

Tactic:
rw [← @comp_leftAdjointMate C, Category.comp_id, @leftAdjointMate_id]

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ Y : C
p₁ : ExactPairing X₁ Y
p₂ : ExactPairing X₂ Y
⊢ 𝟙 ᘁY = 𝟙 X₁
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ Y : C
p₁ : ExactPairing X₁ Y
p₂ : ExactPairing X₂ Y
⊢ ((ᘁ𝟙 Y) ≫ ᘁ𝟙 Y) = 𝟙 X₂

Tactic:
rw [← @comp_leftAdjointMate C, Category.comp_id, @leftAdjointMate_id]

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ Y : C
p₁ : ExactPairing X₁ Y
p₂ : ExactPairing X₂ Y
⊢ 𝟙 ᘁY = 𝟙 X₂
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ Nat.card (M →ₐ[K] L) ≤ finrank K M

Tactic:
convert! toNat_le_toNat (cardinalMk_algHom_le_rank K M L) ?_

Goal state after:
case e'_4
K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ finrank K M = toNat (lift.{v, u_2} (Module.rank K M))

K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ lift.{v, u_2} (Module.rank K M) < ℵ₀
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
this : Nontrivial K
⊢ Module.rank K M < ℵ₀

Tactic:
apply Module.rank_lt_aleph0

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
M : Type u_5
M₁ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : CommSemiring R₁
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R₁ M₁
I₁ I₂ : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M
H : B.IsAlt
⊢ B.IsRefl

Tactic:
intro x

Goal state after:
R : Type u_1
R₁ : Type u_2
M : Type u_5
M₁ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : CommSemiring R₁
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R₁ M₁
I₁ I₂ : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M
H : B.IsAlt
x : M₁
⊢ ∀ (y : M₁), (B x) y = 0 → (B y) x = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mpr
R : Type u_1
R₁ : Type u_2
M₁ : Type u_6
inst✝⁵ : CommRing R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
I : R₁ →+* R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R
h : B = -B.flip
⊢ B.IsAlt

Tactic:
intro x

Goal state after:
case mpr
R : Type u_1
R₁ : Type u_2
M₁ : Type u_6
inst✝⁵ : CommRing R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
I : R₁ →+* R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R
h : B = -B.flip
x : M₁
⊢ (B x) x = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_20
M : Type u_21
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : LinearMap.BilinForm R M
F : Type u_22
inst✝¹ : FunLike F M M
inst✝ : LinearMapClass F R M M
f : F
h : IsLeftRegular 2
hB : IsSymm B
hf : ∀ (x : M), (B (f x)) (f x) = (B x) x
⊢ IsOrthogonal B ⇑f

Tactic:
intro x

Goal state after:
R : Type u_20
M : Type u_21
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : LinearMap.BilinForm R M
F : Type u_22
inst✝¹ : FunLike F M M
inst✝ : LinearMapClass F R M M
f : F
h : IsLeftRegular 2
hB : IsSymm B
hf : ∀ (x : M), (B (f x)) (f x) = (B x) x
x : M
⊢ ∀ (y : M), (B (f x)) (f y) = (B x) y
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid Mₗ₁
inst✝⁶ : AddCommMonoid Mₗ₂
inst✝⁵ : AddCommMonoid Mₗ₁'
inst✝⁴ : AddCommMonoid Mₗ₂'
inst✝³ : Module R Mₗ₁
inst✝² : Module R Mₗ₂
inst✝¹ : Module R Mₗ₁'
inst✝ : Module R Mₗ₂'
B : Mₗ₁ →ₗ[R] Mₗ₂ →ₗ[R] M
e₁ : Mₗ₁ ≃ₗ[R] Mₗ₁'
e₂ : Mₗ₂ ≃ₗ[R] Mₗ₂'
h : B.SeparatingLeft
⊢ ((e₁.arrowCongr (e₂.arrowCongr (LinearEquiv.refl R M))) B).SeparatingLeft

Tactic:
intro x

Goal state after:
R : Type u_1
M : Type u_5
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid Mₗ₁
inst✝⁶ : AddCommMonoid Mₗ₂
inst✝⁵ : AddCommMonoid Mₗ₁'
inst✝⁴ : AddCommMonoid Mₗ₂'
inst✝³ : Module R Mₗ₁
inst✝² : Module R Mₗ₂
inst✝¹ : Module R Mₗ₁'
inst✝ : Module R Mₗ₂'
B : Mₗ₁ →ₗ[R] Mₗ₂ →ₗ[R] M
e₁ : Mₗ₁ ≃ₗ[R] Mₗ₁'
e₂ : Mₗ₂ ≃ₗ[R] Mₗ₂'
h : B.SeparatingLeft
x : Mₗ₁'
⊢ (∀ (y : Mₗ₂'), (((e₁.arrowCongr (e₂.arrowCongr (LinearEquiv.refl R M))) B) x) y = 0) → x = 0
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: case right
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W : Subspace K V₁
⊢ (Submodule.dualPairing W).SeparatingRight

Tactic:
intro x

Goal state after:
case right
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W : Subspace K V₁
x : ↥W
⊢ (∀ (x_1 : Dual K V₁ ⧸ dualAnnihilator W), ((Submodule.dualPairing W) x_1) x = 0) → x = 0
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W : Subspace K V₁
⊢ (dualCopairing W).Nondegenerate

Tactic:
constructor
  · rw [LinearMap.separatingLeft_iff_ker_eq_bot, dualCopairing_eq]
    apply LinearEquiv.ker
  · rintro ⟨x⟩
    simp only [Quotient.quot_mk_eq_mk, dualCopairing_apply, Quotient.mk_eq_zero]
    rw [← forall_mem_dualAnnihilator_apply_eq_zero_iff, SetLike.forall]
    exact id

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hl : IsLimit (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))
Z : C
x : Z ⟶ A
hx : x ≫ r A = 0
hxx : (x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A) = 0
y : Z ⟶ (KernelFork.ofι (diag A) ⋯).pt
hy : y ≫ diag A = x ≫ prod.lift (𝟙 A) 0
⊢ y = 0

Tactic:
← Category.comp_id y,

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hl : IsLimit (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))
Z : C
x : Z ⟶ A
hx : x ≫ r A = 0
hxx : (x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A) = 0
y : Z ⟶ (KernelFork.ofι (diag A) ⋯).pt
hy : y ≫ diag A = x ≫ prod.lift (𝟙 A) 0
⊢ y ≫ 𝟙 (KernelFork.ofι (diag A) ⋯).pt = 0
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0
⊢ ∀ (s : Fork prod.snd 0) (m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt),
    m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι → m = s.ι ≫ prod.fst

Tactic:
intro s m h

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0
s : Fork prod.snd 0
m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt
h : m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι
⊢ m = s.ι ≫ prod.fst
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ 0 - -a = a

Tactic:
neg_def

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ 0 - (0 - a) = a
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ - -a = a

Tactic:
rw [neg_def, neg_def]
  conv_lhs =>
    congr; rw [← sub_self a]
  rw [sub_sub_sub, sub_zero, sub_self, sub_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ b - (0 - a) = b - -a

Tactic:
neg_def

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ b - (0 - a) = b - (0 - a)
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - (a - b) = -a + b

Tactic:
neg_def

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - (a - b) = 0 - a + b
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - a - (0 - b) = 0 - a - -b

Tactic:
neg_def

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - a - (0 - b) = 0 - a - (0 - b)
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ f ≫ g - (0 - f ≫ h) = f ≫ g - -f ≫ h

Tactic:
neg_def

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ f ≫ g - (0 - f ≫ h) = f ≫ g - (0 - f ≫ h)
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ f ≫ h - (0 - g ≫ h) = f ≫ h - -g ≫ h

Tactic:
neg_def

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ f ≫ h - (0 - g ≫ h) = f ≫ h - (0 - g ≫ h)
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : GrpCat
f : A ⟶ B
x : ↑B
hx : x ∈ (Hom.hom f).range
⊢ τ ((g x) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩)) = ∞

Tactic:
g_apply_fromCoset

Goal state after:
A B : GrpCat
f : A ⟶ B
x : ↑B
hx : x ∈ (Hom.hom f).range
⊢ τ (fromCoset ⟨x • ↑⟨↑(Hom.hom f).range, ⋯⟩, ⋯⟩) = ∞
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: case refine_1.fromCoset
A B : GrpCat
f : A ⟶ B
a : ↑A
y : ↑B
⊢ (h ((ConcreteCategory.hom f) a)) (fromCoset ⟨(fun x ↦ x • ↑(Hom.hom f).range) y, ⋯⟩) =
    (g ((ConcreteCategory.hom f) a)) (fromCoset ⟨(fun x ↦ x • ↑(Hom.hom f).range) y, ⋯⟩)

Tactic:
g_apply_fromCoset

Goal state after:
case refine_1.fromCoset
A B : GrpCat
f : A ⟶ B
a : ↑A
y : ↑B
⊢ (h ((ConcreteCategory.hom f) a)) (fromCoset ⟨(fun x ↦ x • ↑(Hom.hom f).range) y, ⋯⟩) =
    fromCoset ⟨(ConcreteCategory.hom f) a • ↑⟨(fun x ↦ x • ↑(Hom.hom f).range) y, ⋯⟩, ⋯⟩
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
h : HomologyMapData φ h₁ h₂
⊢ h.left.φH ≫ h₂.iso.hom = h₁.iso.hom ≫ h.right.φH

Tactic:
by
  simp only [← cancel_epi h₁.left.π, ← cancel_mono h₂.right.ι, assoc,
    LeftHomologyMapData.commπ_assoc, HomologyData.comm, LeftHomologyMapData.commi_assoc,
    RightHomologyMapData.commι, HomologyData.comm_assoc, RightHomologyMapData.commp]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
φK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯
commf' : h₁.f' ≫ φK = φ.τ₁ ≫ h₂.f'
φH : h₁.H ⟶ h₂.H := h₁.descH (φK ≫ h₂.π) ⋯
⊢ h₁.π ≫ φH = φK ≫ h₂.π

Tactic:
by simp [φH]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂
⊢ φ.τ₂ ≫ h₂.p = h₁.p ≫ ψ₂.φQ

Tactic:
commp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂
⊢ φ.τ₂ ≫ h₂.p = φ.τ₂ ≫ h₂.p
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
φH : h₁.H ⟶ h₂.H := h₂.liftH (h₁.ι ≫ φQ) ⋯
⊢ φH ≫ h₂.ι = h₁.ι ≫ φQ

Tactic:
by simp [φH]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ rightHomologyMap φ ≫ h₂.rightHomologyIso.hom = h₁.rightHomologyIso.hom ≫ γ.φH

Tactic:
by
  simp only [γ.rightHomologyMap_eq, assoc, Iso.inv_hom_id, comp_id]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ (h₂.i ≫ inv φ.τ₂) ≫ S₁.g = 0

pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' φ h₁ h₂ ≫ h₁.liftK (h₂.i ≫ inv φ.τ₂) ?refine_1 = 𝟙 h₁.K

pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ h₁.liftK (h₂.i ≫ inv φ.τ₂) ?refine_1 ≫ cyclesMap' φ h₁ h₂ = 𝟙 h₂.K

Tactic:
· simp only [assoc, ← cancel_mono φ.τ₃, zero_comp, ← φ.comm₂₃, IsIso.inv_hom_id_assoc, h₂.wi]

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' φ h₁ h₂ ≫ h₁.liftK (h₂.i ≫ inv φ.τ₂) ⋯ = 𝟙 h₁.K

case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ h₁.liftK (h₂.i ≫ inv φ.τ₂) ⋯ ≫ cyclesMap' φ h₁ h₂ = 𝟙 h₂.K
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ⋯ ≫ opcyclesMap' φ h₁ h₂ = 𝟙 h₂.Q

Tactic:
simp only [← cancel_epi h₂.p, h₂.p_descQ_assoc, assoc, p_opcyclesMap',
      IsIso.inv_hom_id_assoc, comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
⊢ (if h : (ConcreteCategory.hom f) y ∈ Set.range ⇑(ConcreteCategory.hom f) then
      (ConcreteCategory.hom g) (Classical.choose h)
    else ⋯.some) =
    (ConcreteCategory.hom g).toFun y

Tactic:
split_ifs <;> rename_i h

Goal state after:
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h : (ConcreteCategory.hom f) y ∈ Set.range ⇑(ConcreteCategory.hom f)
⊢ (ConcreteCategory.hom g) (Classical.choose h) = (ConcreteCategory.hom g).toFun y

case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h : (ConcreteCategory.hom f) y ∉ Set.range ⇑(ConcreteCategory.hom f)
⊢ ⋯.some = (ConcreteCategory.hom g).toFun y
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ

Tactic:
rw [γ.quasiIso_iff, (γ.map F).quasiIso_iff, RightHomologyMapData.map_φH]

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH
Module: Mathlib.CategoryTheory.Subobject.MonoOver

Goal state before:
pp: case of
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type u₃
inst✝ : Category.{v₃, u₃} J
X : C
F : Over X
x✝ : (Over.isMono X).limitsOfShape J F
p : LimitPresentation J F
hp : ∀ (j : J), Over.isMono X (p.diag.obj j)
Z✝ : C
g h : Z✝ ⟶ F.left
e : g ≫ F.hom = h ≫ F.hom
j : J
⊢ g ≫ (WithTerminal.coneEquiv.functor.obj { pt := F, π := p.π }).π.app (WithTerminal.of j) =
    h ≫ (WithTerminal.coneEquiv.functor.obj { pt := F, π := p.π }).π.app (WithTerminal.of j)

Tactic:
| of j =>

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : Type u₁
inst✝⁵ : Category.{v₁, u₁} J
K : Type u₂
inst✝⁴ : Category.{v₂, u₂} K
D : Type u₃
inst✝³ : Category.{v₃, u₃} D
inst✝² : HasLimitsOfShape J C
inst✝¹ : HasColimitsOfShape K C
inst✝ : PreservesLimitsOfShape J colim
d : D
X : K ⥤ D ⥤ C
⊢ colimit (X ⋙ (evaluation D C).obj d) ≅
    colimit (((whiskeringRight K (D ⥤ C) C).obj ((evaluation D C).obj d)).flip ⋙ (evaluation (K ⥤ D ⥤ C) C).obj X)

Tactic:
exact HasColimit.isoOfNatIso (Iso.refl _)

Goal state after: